We are actually interested in \(\left(\varepsilon ,\varepsilon ^{\prime } \right) =\left( -,-\right)\)  for \(n/2\)  even. Note that, \(O_{n/2}^{-}(q^{2}).Z_{2}\)  preserves \(\mathcal {S}\)  by [1]}, Proposition 2.4. Then \(X_{\mathcal {S}} \cong P\Omega _{n/2}^{-}(q^{2}).Z_{2}\)  by [1]}, Main Theorem (b), and by [3]}, Proposition 4.3.16.(II). We use these information to prove the following lemma, which will play a central role in excluding some of the cases where \(X_{x}\)  is a \(\mathcal {C}_{3}\) -subgroup of \(X\) .
